Athermal Operation of Silicon Waveguides: Spectral,
Second Order and Footprint Dependencies
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Citation
Vivek Raghunathan, Winnie N. Ye, Juejun Hu, Tomoyuki
Izuhara, Jurgen Michel, and Lionel Kimerling, "Athermal
operation of Silicon waveguides: spectral, second order and
footprint dependencies," Opt. Express 18, 17631-17639 (2010)
As Published
http://dx.doi.org/10.1364/OE.18.017631
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Optical Society of America
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Author's final manuscript
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Wed May 25 23:13:38 EDT 2016
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http://hdl.handle.net/1721.1/58117
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Athermal operation of Silicon waveguides:
spectral, second order and footprint
dependencies
Vivek Raghunathan,1 Winnie N.Ye,2 Juejun Hu,1 Tomoyuki Izuhara, 3 Jurgen Michel,1
and Lionel Kimerling1
1
Microphotonics Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada
3
Enablence Technologies, 100 Fordham Road, Wilmington, MA-01887
*Corresponding authors: vivekr@mit.edu, lckim@mit.edu
Abstract: We report the design criteria and performance of Si ring
resonators for passive athermal applications in wavelength division
multiplexing (WDM). The waveguide design rules address i) positivenegative thermo-optic (TO) composite structures, ii) resonant wavelength
dependent geometry to achieve constant confinement factor ( Γ ), and iii)
observation of small residual second order effects. We develop exact design
requirements for a temperature dependent resonant wavelength shift
(TDWS) of 0 pm/K and present prototype TDWS performance of 0.5pm/K.
We evaluate the materials selection tradeoffs between high-index contrast
(HIC) and low-index contrast (LIC) systems and show, remarkably, that
FSR and footprint become comparable under the constraint of athermal
design.
©2010 Optical Society of America
OCIS codes: (130.2790) Guided waves; (130.3130) Integrated optics materials; (160.6840)
Thermo-Optical Materials; (160.5470) Polymers; (230.5750) Resonators; (230.7370)
Waveguides.
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1. Introduction
Silicon is a compelling material choice for electronic and photonic applications using CMOS
fabrication technology and requiring seamless electronic-photonic integration. The resulting
performance benefits of improved power efficiency, increased functionality, enhanced
reliability, and reduced cost provide strong incentives for on chip electronic-photonic
integration. However, high levels of device integration create significant local and global chip
temperature excursions during operation that prove problematic for silicon-based photonics
due to the high thermo-optic coefficient of silicon. In particular, thermo-optic (TO) index
variations will induce temperature dependent wavelength shifts (TDWS) of resonant devices
that limit wavelength resolution in applications such as wavelength division multiplexing
(WDM) and high-resolution spectroscopy. Active thermal compensation by heaters and
thermo-electric coolers are the legacy solution for low-density integration. In fact, the silicon
CMOS layered architecture was shown recently to be ideal for embedded thermo-optic phase
shifters for tuning complex filters [1]. However, the required electrical power and the number
of I/O lines ultimately limit integration density. Passive thermo-optic compensation of silica
photonic devices by a negative thermo-optic polymer cladding (PMMA) was first reported by
Kokubun et al. [2-4]. Lee et al. [5,6] have extended the application of compensating polymer
claddings to silicon ring resonators and have exhibited significant thermo-optic compensation
to a TDWS of 2pm/K. Teng et al. [7] used a similar approach for SOI rings with a polymer
top-cladding. Guha et al. [8] and Huang et al. [9] have employed alternate device design
solutions to overcome the thermal dependence in interference-based devices. We have
recently shown that exact athermal design is achievable for both symmetric and asymmetric
waveguide-cladding geometries [10]. This communication reports the application of those
design principles with the creation of prototypes that exhibit reproducible TDWS = 0.5pm/K
for silicon-on-oxide ring resonators and reveal, with these devices, ultimate spectral, secondorder and footprint constraints for athermal silicon waveguides.
2. Theory
2.1 Polymer over-cladding
The effective index (neff) of a waveguide system depends on the refractive index of the core
(nc) and the cladding (ncl). For an asymmetric channel waveguide system, with a different top
and bottom cladding, we can express the system thermo-optic coefficient [10,11] as
dneff
dT
(λ ) = Γ c (λ )
dnc
dn
dn
(λ ) + Γtcl (λ ) tcl (λ ) + Γbcl (λ ) bcl (λ )
dT
dT
dT
(1)
where dneff (λ) is the effective TO coefficient of the waveguide system at a given wavelength,
dT
while dnc (λ ) ,
dT
dn tcl
and dn bcl ( λ )
(λ )
dT
dT
represent the TO coefficients of the core, the top cladding
and the bottom cladding respectively. Parameters Γc (λ) , Γtcl (λ ) and Γbcl (λ ) in the above
equation represent the confinement factor of the core, the top and bottom cladding
respectively as defined by Robinson et al.[11]. Henceforth, in all of our formulations, we shall
consider only the effect of top cladding on the TO coefficient of the system. This assumption
is justified in the case of SiO2 as the bottom cladding, since its TO coefficient is an order of
magnitude less than that of the Si waveguide and the polymer top clad as shown in Table 1.
Eq.(1) is a first order approximation of the refractive index variation with temperature where
we ignore the variation of Γ(λ) with temperature and the second order variation of the
refractive index of the core and cladding. We assume that the thermal expansion coefficient of
the substrate doesn’t change the geometry of the channel waveguide and hence the
confinement factor in the given temperature range. Also note that we have assumed that the
thermo-optic coefficient of the core and the cladding do not vary with wavelengths, since a-Si,
SiO2, and polymers do not have any dispersion in the wavelength near the IR region.
Athermal operation of a waveguide system is defined by the condition of dneff (λ ) = 0 ,
dT
which from Eq.(1) can be reached only if the cladding has a negative thermo-optic coefficient.
Motivated by this theory, we use polymer cladding for Si waveguides, as polymers are known
to have a negative thermo-optic coefficient. The high thermo-optic coefficient of a-Si (Table
1) requires selection of polymers with an equally high thermo-optic coefficient. Many
commercially available polymers have TO coefficients [12] smaller than a-Si. We use a
proprietary hyperlinked fluoropolymer recently developed by Enablence (EP) that has been
designed to have a high dn/dT = -2.65×10-4 K-1 and a refractive index of 1.3793 at 1550nm.
Table 1. Refractive index and TO coefficient of materials in this study
Material
a-Si
SiO2
Si3N4
EP
n (@1550nm)
3.48
1.46
2.05
1.3793
dn/dT (×10-4 K-1)
2.3
0.1
0.4
-2.65
2.2 Wavelength dependence and second order effects
In Eq. (1), note the dependence of dneff (λ ) on wavelength arising due to the variation of
dT
confinement factor with wavelength. In other words, we would expect Γ(λ) to decrease with
increasing wavelength owing to the longer evanescent tail. This dependence proves even more
crucial for waveguides of dimensions 700nm×216nm, as we are studying wavelengths in the
near IR regime (1450nm -1550nm) whose magnitude in the core is comparable to the core
dimension. The single mode confinement required for ring resonators carries the important
consequence that athermal operation can be achieved at only one wavelength for a given
waveguide dimension.
The variation of confinement factor with wavelength influences the effective TO
coefficient, which in turn manifests as a dependence of TDWS ( d λ r ) on wavelength. TDWS
dT
can be related to the effective TO coefficient using Eq.(2).
1 d λr
1 dneff
(λ ) =
(λ )
λr dT
ng dT
(2)
It is more practical to simulate TDWS, as opposed to the effective system TO coefficient,
because TDWS is the experimental observable. Furthermore, fine wavelength resolution
(roughly 1pm) and hence very high refractive index measurement accuracy may be achieved
through a resonant peak fitting algorithm [13].
Our model predicts a decrease in effective thermo-optic coefficient with increasing
resonant wavelength due to the resulting decrease of Γ(λ) .As the propagating wavelength
increases at constant waveguide dimension, the effective TO contribution from the cladding
increases. The slope (rate of TO change) for the polymer clad structure is higher than for the
the oxide clad structure, because the magnitude of the thermo-optic coefficient of polymer EP
is higher than for the oxide. It is apparent from Eq. (1) that the slope will be larger for the EP
polymer cladding as the contribution from the second term becomes significant. For a TM
mode, Γ varies significantly by changing the height of the waveguide. With an appropriate
polymer cladding, a true athermal operation (i.e. TDWS=0) can be achieved for a thin
waveguide due to the reduced confinement and greater expansion of the mode into the
cladding regime.
Any residual second order terms should become significant when the first order terms
vanish according to dneff (λ ) = 0 at very small resonance shifts (<2pm/K). It is a reasonable
dT
assumption that the resonant wavelength shifts linearly with temperature, and it is largely
governed by Eq. (1) and Eq. (2). Such linear relations hold true for relatively large peak shifts
(>10pm/K). However at extremely small wavelength shifts, the second order contributions
arising from the variation of Γ with temperature and possibly the second order contribution
from the refractive index become significant and cannot be neglected. In theory, with increase
in temperature, the refractive index of the core increases and that of the polymer cladding
decreases, thereby increasing the confinement factor resulting in a larger peak shift. Thus the
refractive index variation including the second order term can be written as follows
∂nc
∂n
+Γcl ⋅ cl ) ⋅ dT
∂T
∂T
(3)
2
∂ nc
∂2ncl ∂Γc ∂nc 2 ∂Γcl ∂ncl 2 ∂Γc ∂nc ∂ncl ∂Γcl ∂nc ∂ncl
1
2
+ ⋅[Γc 2 +Γcl 2 +
+
( ) +
( ) +
]⋅ dT
∂T
∂T ∂nc ∂T
∂ncl ∂T
∂ncl ∂T ∂T ∂nc ∂T ∂T
2
dneff (nc , ncl ) = (Γc ⋅
where Γc and Γcl are the confinement factor of the core and the cladding (only the polymer
top cladding is considered), respectively. The second order terms can be categorized into two
types of contributions:
2
2
1) the term 1 ⋅ ( Γc ∂ nc + Γcl ∂ ncl ) corresponds to the second order TO coefficients of the core
2
2
∂T
∂T 2
and cladding materials;
2) the term 1 ⋅ [ ∂Γc ( ∂nc ) 2 + ∂Γcl ( ∂ncl ) 2 + ∂Γc ∂nc ∂ncl + ∂Γcl ∂nc ∂ncl ] represents the TDWS
∂ncl ∂T
∂ncl ∂T ∂T
∂nc ∂T ∂T
2 ∂nc ∂T
due to the variation of confinement factors at different temperatures.
2.3 Design and fabrication constraints: TE mode vs TM mode
The design leverage of the passive approach outlined above is that athermal operation is
achieved universally for a particular confinement factor resulting from a selected corepolymer cladding system (Eq. (1)). This confinement factor is however reached for different
waveguide dimensions for TE and TM modes. For example, reduction of the confinement
factor is achieved by reducing the waveguide width for TE mode and by reducing its height
for TM mode. Thus, without any loss of generality, one can safely claim that the physics
behind the behavior of TE and TM mode is similar as long as one tailors the waveguide
design to have the same confinement factor for each mode. The proof-of-concept prototypes
in the experimental section of this paper to follow are designed for exclusive TM mode
propagation. The lithographic constraints of our fabrication facility allow a minimum
waveguide width of 700nm and adequate mode penetration into the upper polymer cladding is
accomplished by the use of a-Si waveguides of reduced thickness. The a-Si ring resonators
have racetrack configurations with radii of 20 microns, where the bending loss (7.84×10-4
dB/cm) of TM mode is insignificant. Furthermore, the loss of these devices is largely
governed by the scattering loss which is smaller for the TM mode than the TE mode. Owing
to the high loss and low evanescent coupling coefficient associated with the TE mode, we do
not resolve resonant peaks in the TE polarization transmission spectra, and the presented
performance data are for the TM polarization. We demonstrate that TM mode propagation
can achieve athermal operation at relatively wider waveguide widths and smaller thickness.
However, the governing theory and the design rules apply equally well for TE mode.
Fig.1. (a) Schematic view of an a-Si racetrack resonator and (b) the transmission spectra
showing the resonance peak shift with temperature for both oxide clad and polymer clad
devices. We compare a-Si rings of the same cross-section (700nm × 216nm) and different top
claddings (oxide and polymer). In the case of an oxide clad ring, the positive TO coefficient of
both core and cladding result in a positive peak shift of 63pm/K. There is significant TO
compensation for the polymer clad ring and the peak shift is reduced to 7pm/K for this
waveguide design.
3. Prototype fabrication
We fabricated a-Si ring resonators with a racetrack configuration (Fig. (1a)) via
photolithography using an i-line stepper. We deposited two sets of a-Si films, one with
thickness of 206nm and other with 216nm. The actual dimensions of the waveguides are
measured to be 700nm×216nm and 700nm×206nm. Both waveguides are designed to operate
at single mode for TM polarization. We deposited a CVD (chemical vapor deposition) oxide
film on one set of waveguides to be used as a reference; while the other sets of devices were
spun with a hyperlinked fluoropolymer sample, EP, at Enablence Inc. The spin coating
process was followed by UV exposure after which the samples are baked in vacuum for
around 8 hrs to cross-link the EP polymer. We then measure the TM mode transmission
spectrum at various temperatures to monitor the resonant peak positions and their thermooptic induced changes.
4. Prototype performance
Fig. (1b) shows the transmission spectra comparing the performance of oxide over-cladding
and EP over-cladding at various temperatures. The glass transition temperature of EP lies
below room temperature and hence outside the operating range of our interest (25-125oC). We
thus neglect the higher order variations due to the polymer glass transition in our experiments.
We measure the resonant wavelength shift for the oxide and polymer clad waveguides over a
range of temperatures as is shown in Fig. (2).
Fig. 2. Experimental results showing dλr / dT variation with
λr for TM mode propagation. The
slope of the dλr / dT variation with λr reflects the role of Γ (see Eqn. (1), (2)) on TO and
dλr / dT . Hence the wavelength at the dashed-line crossover point is an unique condition for
dλr / dT = 0. The hexagonal bold data points correspond to the spectral data shown in Fig. (1)
Fig. (2) expresses the effects of both thermo-optic coefficient of the cladding and
confinement factor on the resonance shift. The measurements for each resonant wavelength
were carried out between 25oC-60oC and every data point is a linear regression of 8 data
points taken at every 5oC interval. For large resonance shifts (TDWS > 10pm/K), R2 ≥ 0.99,
and for near athermal performance (TDWS ≤ 1pm/K) R2 ≥ 0.94 due to reduced linear
correlation arising from the residual second order terms discussed below. The dramatic
decrease of TDWS from 65 pm/K for an oxide cladding to as low as 0.5pm/K at 1524nm for
the EP polymer cladding can be attributed to the negative thermo-optic coefficient of the
polymer and its effect on the effective thermo-optic coefficient of the system. Figure 2
demonstrates two important results: i) the achievement near perfect thermo-optic
compensation, and ii) a variation of TDWS with wavelength that correlates well with
simulated values. As explained earlier and as verified by the coincidence between data and
simulation, the variation in confinement factor with wavelength changes the TDWS. The
excellent agreement between experimental data and the simulations clearly validates our
theoretical formulation of TDWS in Section 2.
Fig. 3. Variation of athermal waveguide width with wavelength for TE and TM modes. These
simulation results are for an a-Si waveguide of 206nm height with an SiO2 underclad and the
EP polymer over-cladding. The desired waveguide width for athermal operation increases with
wavelength to keep the confinement factor constant, consistent with the theory presented in
section 2.2.
The wavelength dependence of TDWS imposes an additional design constraint of different
waveguide dimensions for different resonant wavelengths to achieve athermal operation. For a
single WDM implementation, the mask design should incorporate varying waveguide widths,
for instance, when the height of the waveguide (layer thickness) is fixed. Figure 3 plots the
range of the wavelength dependent variation in waveguide dimensions for athermal operation
of both TE and TM modes. The simulation assumes an a-Si ring with an oxide undercladding and a polymer over-cladding. We fix the height of the a-Si waveguide at 206nm
and calculate the desired width, to achieve a peak shift of less than 0.5pm/K, at various
wavelengths. The desired width varies from 550nm (at 1500nm) to 1150nm (at 1580nm) as
the wavelength increases. Such a large variation in waveguide width can be correlated to the
small dependence of the confinement factor of a TM mode on width compared to height
variation. Conversely, one can expect to achieve athermal operation of a TE mode over a
smaller range of widths for a given range of wavelengths. Fig. (3) clearly shows this stronger
dependence of confinement factor of a TE mode on the width of the waveguide. Fig. (3) also
defines the ‘athermal constraint’ on fabrication tolerance, since changes in the waveguide
width can shift athermal operation to a different wavelength. As an example, an error of
100nm in the waveguide width would shift the “athermal response peak” by 10nm.
Residual second order variations are revealed when TDWS is small. Fig. (4) shows the
experimental variation of resonant wavelength with temperature for a 700nm × 206nm a-Si
waveguide with an EP cladding.
Fig. 4. Measured second order variations of resonant wavelength with fit to a quadratic
dependence. The residual second order effects become important when the first order terms
vanish at low peak shifts (<2pm/K). The quadratic term has contributions from the temperature
dependence of the confinement factor and from the second order material coefficients.
The residual quadratic term seen in experiments (9×10-5±1×10-5nmK-2) is of the same
order of magnitude predicted in the simulation (2×10-5 ± 1×10-6 nmK-2). The higher value of
the experimental second order term compared to simulation suggests a significant contribution
2
2
from the second order material coefficients, ∂ nc and ∂ ncl , that we neglected in the
∂T 2
∂T 2
simulations (first 2 second order terms in Eq. (3)). The first 2 second order terms of Eq. (3)
can be calculated based on our the experimentally measured second order term, and a value of
4×10-7 ± 1×10-7K-2 for the second order material coefficient of the waveguide system (in other
words, second order effective material TO coefficient) results. Cocorullo et al. [14] have
reported that the second order thermo-optic coefficient of a-Si is roughly 1.43×10-6K-2, so the
determined value is in a reasonable range for the polymer clad system.
5. Material selection for high density integration: Si rings vs Si3N4 rings
The motivation behind using Si ring resonators for WDM filter applications arises from its
high-index contrast and high mode confinement. As a result, Si rings can have small bending
radii resulting in large FSR. However, athermal operation of these rings requires reduced
confinement (Γ=0.58) for the mode to penetrate the cladding. The result is a lower effective
index (1.78) waveguide with higher radiative bending loss. This design rule leads one to
inquire whether a lower-index contrast (LIC) waveguide core material with a lower TO
coefficient, such as Si3N4 (TO=4×10-5), would perform as well or better for athermal WDM
applications. We use bend radius (Rb) as a variable in the figure of merit for the ring
resonator, and we calculate for TM mode to be consistent with our experimental results. We
start the comparison by finding the confinement factor required for athermal operation of
nitride rings with the same polymer cladding used for a-Si rings. We then proceed to find the
bending radius for both Si and Si3N4 for a given bending loss [15], which we fixed at
4.34dB/cm (or 1cm-1).
Table 2 summarizes Rb for both systems. As the results show, the advantage of high-index
contrast (HIC) (a-Si) is neutralized under the constraint of athermal operation, and the Rb
performance is similar to the lower-index Si3N4 core. Therefore, materials selection of lower
index contrast systems like Si3N4 might prove advantageous, since the lower TO coefficient
may allow the use of a wider range of commercially available polymers to achieve athermal
behavior. Si3N4 rings have a higher FSR due to their smaller group index (ng), and they could
prove additionally useful for athermal WDM applications where FSR is the primary
performance driver. When developing scaling rules for the material selection for athermal
performance, it is important to realize that HIC systems lose their benefits over LIC systems
in terms of small Rb and high FSR, and the two performances becomes comparable.
Table 2. Performance comparison of two athermal waveguide designs, for an a-Si core and a Si3N4 core, with the
same EP polymer overclad
Material
n
Γ
neff
ng
Rb(microns)
Si
3.48
0.58
1.78
3.68
3.5
Si3N4
2.05
0.9
1.75
2.2
5.5
6. Summary and conclusion
We develop the design principles for athermal waveguide device performance. We
demonstrate prototype ring resonators with near complete thermo-optic compensation of an
a-Si on SiO2 waveguide using a polymer cladding. We measure a resonant wavelength shift of
0.5pm/K, which is the lowest TDWS value reported to date for silicon resonators. The design
formalism and prototypes establish the requirement for a unique waveguide dimension for
each operating wavelength. The athermal resonator performance reveals residual second order
terms that are also described in the theoretical section. This report validates a general solution
for temperature-independent performance of silicon resonant photonic devices. An evaluation
of the performance tradeoffs of athermal HIC systems in terms of FSR and bend radius shows
that LIC systems might be superior to HIC systems in certain applications.
Acknowledgement
We acknowledge Dr. Ningning Feng of Kotura for support with simulations for bending loss
calculations. This work is sponsored under the Defense Advanced Research Projects
Agency’s Athermal Photonic Circuits (APhoCs) program. The program is executed by the
Microsystems Technology Office (MTO) under Award No. W911NF-09-1-0059, Program
manager: Mike Haney.